Exact Algorithms for No-Rainbow Coloring and Phylogenetic Decisiveness
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The input to the no-rainbow hypergraph coloring problem is a hypergraph H where every hyperedge has r nodes. The question is whether there exists an r-coloring of the nodes of H such that all r colors are used and there is no rainbow hyperedge – i.e., no hyperedge uses all r colors. The no-rainbow hypergraph r-coloring problem is known to be NP-complete for r ≥ 3. The special case of r=4 is the complement of the phylogenetic decisiveness problem. Here we present a deterministic algorithm that solves the no-rainbow r-coloring problem in O^*((r-1)^(r-1)n/r) time and a randomized algorithm that solves the problem in O^*((r/2)^n) time.
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